Topological Recursion: Introduction, Overview and Applications

I will give a talk about the topological recursion (TR) of Eynard and Orantin, which generates from some initial data (the so-called the spectral curve) a family of symmetric multi-differentials on a Riemann surface. Symplectic transformations of the spectral curve play an important role and are conjectured to leave the free energies $F_g$ invariant. TR has nowadays a lot of applications ranging random matrix theory, integrable systems, intersection theory on the moduli space of complex curves $\mathcal{M}_{g,n}$, topological string theory over knot theory to free probability theory. I will highlight specific examples, such as the Airy curve (also sometimes called the Kontsevich-Witten curve) which enumerates $\psi$-class intersection numbers on $\mathcal{M}_{g,n}$, the Mirzakhani curve for computing Weil–Petersson volumes, the spectral curve of the hermitian 1-matrix model, and the topological vertex curve which derives the $B$-model correlators in topological string theory. Should time allow, I will also discuss the quantum spectral curve as a quantisation of the classical spectral curve annihilating a wave function constructed from the family of multi-differentials.